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            <h1 style="display: none">algorithm-基础算法</h1>
            
              <p class="note note-info">
                
                  本文最后更新于：7 个月前
                
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            <div class="markdown-body" id="post-body">
              <h2 id="0-x-前言"><a href="#0-x-前言" class="headerlink" title="[0.x]前言"></a>[0.x]前言</h2><p>参考书目：<strong>《算法竞赛进阶指导》</strong></p>
<p>语言：$Cpp$</p>
<a id="more"></a>
<h2 id="目录"><a href="#目录" class="headerlink" title="目录"></a>目录</h2><ul>
<li>[0.x]前言</li>
<li>[1.x]位运算</li>
<li>[2.x]递推与递归</li>
<li>[3.x]前缀和与差分</li>
<li><strong>[4.x]二分</strong></li>
<li>[5.x]排序</li>
<li><strong>[6.x]倍增</strong></li>
<li><strong>[7.x]贪心</strong></li>
</ul>
<hr>
<h2 id="1-x-位运算"><a href="#1-x-位运算" class="headerlink" title="[1.x]位运算"></a>[1.x]位运算</h2><div class="table-container">
<table>
<thead>
<tr>
<th>与(&amp;)</th>
<th>或(丨)</th>
<th>非(!)</th>
<th>亦或(^)</th>
</tr>
</thead>
<tbody>
<tr>
<td>同1才1</td>
<td>有1即1</td>
<td>反</td>
<td>相同为0，不同为1</td>
</tr>
</tbody>
</table>
</div>
<script type="math/tex; mode=display">
1 << n = 2^n</script><script type="math/tex; mode=display">
n << 1 = 2*n</script><h3 id="快速幂"><a href="#快速幂" class="headerlink" title="快速幂"></a>快速幂</h3><div class="hljs"><pre><code class="hljs cpp"><span class="hljs-comment">//位运算实现快速幂</span>
<span class="hljs-function">ll <span class="hljs-title">quick_pow</span><span class="hljs-params">(ll x,ll p)</span></span>&#123;<span class="hljs-comment">//x^p</span>
    ll base = x,ans = <span class="hljs-number">1</span>;
    <span class="hljs-keyword">while</span>(p)&#123;
        <span class="hljs-keyword">if</span>(p &amp; <span class="hljs-number">1</span>) ans *= base;
        <span class="hljs-comment">// p &amp; 1 = true 时说明p的第一位是1，x</span>
        base *= base , p &gt;&gt;= <span class="hljs-number">1</span>;
        <span class="hljs-comment">// p &gt;&gt;= 1 是右移运算，基本上类似于 p /= 2 ，但是效率更高</span>
    &#125;
    <span class="hljs-keyword">return</span> ans;
&#125;
<span class="hljs-comment">//更改乘法方案可以实现矩阵加速</span>
<span class="hljs-function">ll <span class="hljs-title">m_pow</span><span class="hljs-params">(node m,ll p)</span></span>&#123;<span class="hljs-comment">//矩阵m^p</span>
    node base = x,ans = e;<span class="hljs-comment">//e为单位矩阵</span>
    <span class="hljs-keyword">while</span>(p)&#123;
        <span class="hljs-keyword">if</span>(p &amp; <span class="hljs-number">1</span>) ans = mutiply(ans,base);
        base = mutiply(base,base) , p &gt;&gt;= <span class="hljs-number">1</span>;
    &#125;
    <span class="hljs-keyword">return</span> ans;
&#125;</code></pre></div>
<p><a href="[https://github.com/WeepingDemon/-OI-life-/blob/master/P1939%20%E3%80%90%E6%A8%A1%E6%9D%BF%E3%80%91%E7%9F%A9%E9%98%B5%E5%8A%A0%E9%80%9F.cpp](https://github.com/WeepingDemon/-OI-life-/blob/master/P1939 [模板]矩阵加速.cpp">矩阵快速幂模板</a>)</p>
<p>提示：使用<strong>位运算</strong>请<strong>疯狂</strong>加<strong>括号</strong></p>
<p>如果 $n$ 为偶数，有$n$  $xor$  $1=n+1$<br>如果 $n$ 为奇数，有$n$  $xor$  $1=n-1$<br>即存在：<br><div class="hljs"><pre><code class="hljs cpp"><span class="hljs-number">0</span>^<span class="hljs-number">1</span> = <span class="hljs-number">1</span>   <span class="hljs-number">1</span>^<span class="hljs-number">1</span> = <span class="hljs-number">0</span>
<span class="hljs-number">2</span>^<span class="hljs-number">1</span> = <span class="hljs-number">3</span>   <span class="hljs-number">3</span>^<span class="hljs-number">1</span> = <span class="hljs-number">2</span>
...</code></pre></div></p>
<h3 id="lowbit"><a href="#lowbit" class="headerlink" title="lowbit"></a>lowbit</h3><div class="hljs"><pre><code class="hljs cpp"><span class="hljs-comment">/*</span>
<span class="hljs-comment">lowbit计算二进制表示下的x中最后低位的1即其后的0构成的竖着</span>
<span class="hljs-comment">例如对于二进制数10100B (20D)</span>
<span class="hljs-comment">lowbit(10100B) = 100B = 4D</span>
<span class="hljs-comment">*/</span>
<span class="hljs-function"><span class="hljs-keyword">int</span> <span class="hljs-title">lowbit</span><span class="hljs-params">(<span class="hljs-keyword">int</span> x)</span></span>&#123;
    <span class="hljs-keyword">return</span> x &amp; (-x);
&#125;
<span class="hljs-comment">//树状数组会用到lowbit</span></code></pre></div>
<h2 id="2-x-递推与递归"><a href="#2-x-递推与递归" class="headerlink" title="[2.x]递推与递归"></a>[2.x]递推与递归</h2><ul>
<li>递推 - 正向推导</li>
<li>递归 - 反向回溯</li>
</ul>
<p>需要每一步相似</p>
<div class="table-container">
<table>
<thead>
<tr>
<th>枚举形式</th>
<th>状态空间规模</th>
<th>一般遍历方式</th>
</tr>
</thead>
<tbody>
<tr>
<td>多项式</td>
<td>$n^k$</td>
<td>循环/递推</td>
</tr>
<tr>
<td>指数</td>
<td>$k^n$</td>
<td>递归/位运算</td>
</tr>
<tr>
<td>排列</td>
<td>$n!$</td>
<td>递归/<code>next_permutation()</code></td>
</tr>
<tr>
<td>组合</td>
<td>$C^n_m$</td>
<td>递归/剪枝</td>
</tr>
</tbody>
</table>
</div>
<h2 id="3-x-前缀和与差分"><a href="#3-x-前缀和与差分" class="headerlink" title="[3.x]前缀和与差分"></a>[3.x]前缀和与差分</h2><h3 id="3-1-前缀和"><a href="#3-1-前缀和" class="headerlink" title="[3.1]前缀和"></a>[3.1]前缀和</h3><ul>
<li>前缀和实现$O(1)$查询<strong>区间和</strong></li>
</ul>
<p>预处理：</p>
<script type="math/tex; mode=display">
sum[i] = \sum^{i}_{j=1} a[j]</script><p>$O(1)$查询区间和：</p>
<script type="math/tex; mode=display">
Sum(l,r) = \sum^{r}_{j=l} a[j] = sum[r]-sum[l-1]</script><p><strong>注意</strong>：使用二维前缀和请利用容斥原理，不要忘记减去重合的部分</p>
<h3 id="3-2-差分"><a href="#3-2-差分" class="headerlink" title="[3.2]差分"></a>[3.2]差分</h3><ul>
<li>可以把差分当做<strong>前缀和</strong>的<strong>逆运算</strong></li>
<li>差分实现$O(k+n)$的<strong>区间加法</strong>($k$为<strong>加法次数</strong>，$n$为区间<strong>总长度</strong>)</li>
</ul>
<p>给定数列A[i]，则其<strong>差分数列</strong>B[i]即为：</p>
<script type="math/tex; mode=display">
B[1] = A[1] , B[i] = A[i]-A[i-1](2\leqslant  i \leqslant  n)</script><p>存在以下<strong>性质</strong>：</p>
<p>对$A[l~r]$进行区间加$x$，等效于$B[l]+x$，$B[r+1]-x$后对差分数列求前缀和。</p>
<div class="hljs"><pre><code class="hljs cpp"><span class="hljs-keyword">for</span>(<span class="hljs-keyword">int</span> i = l;i &lt;= r;i++)&#123;
        A[i] += x;
    &#125;<span class="hljs-comment">//朴素区间加法</span>

B[l] += x , B[r+<span class="hljs-number">1</span>] -= x;
<span class="hljs-keyword">for</span>(<span class="hljs-keyword">int</span> i = <span class="hljs-number">1</span>;i &lt;= n;i++)&#123;
        B[i] += B[i<span class="hljs-number">-1</span>];
    &#125;<span class="hljs-comment">//差分区间加法</span></code></pre></div>
<p>在进行多次区间加法操作的时候，差分区间加法的优势就越来越明显。<br>特别的，对于形式为<strong>常数列</strong>的<strong>差分数列</strong>进行<strong>前缀和</strong>可以得到一个<strong>等差数列</strong>。</p>
<h2 id="4-x-二分"><a href="#4-x-二分" class="headerlink" title="[4.x]二分"></a>[4.x]二分</h2><ul>
<li>二分区间<ul>
<li>对于整数域上的区间二分，可以以$l&lt;r$作为循环条件</li>
<li>对于实数域上的区间二分，可以以$l+eps&lt;r$作为循环条件，$eps$一般取$10^{-(k+2)}$，</li>
</ul>
</li>
<li>二分答案</li>
<li>三分<ul>
<li>取左$\frac{1}{3}$点$lmid$和右$\frac{1}{3}$点$rmid$，判断两者相应的函数值，根据答案变化的单调性进行区间缩小，只适用于单峰函数。</li>
</ul>
</li>
</ul>
<h2 id="5-x-排序"><a href="#5-x-排序" class="headerlink" title="[5.x]排序"></a>[5.x]排序</h2><ul>
<li>关于排序的介绍请浏览<a target="_blank" rel="noopener" href="https://weepingdemon.gitee.io/blog/2019/10/17/algSORT/">此处</a></li>
<li>此处仅介绍排序应用</li>
</ul>
<h3 id="5-1-离散化"><a href="#5-1-离散化" class="headerlink" title="[5.1]离散化"></a>[5.1]离散化</h3><p>先对数据进行$sort$，然后用数组对其进行映射，<code>lower_bound(a+1,a+n+1,x)-a</code>查询映射下标。</p>
<h3 id="5-2-中位数"><a href="#5-2-中位数" class="headerlink" title="[5.2]中位数"></a>[5.2]中位数</h3><p>排序后的中位数可能有特殊性质。</p>
<h2 id="6-x-倍增"><a href="#6-x-倍增" class="headerlink" title="[6.x]倍增"></a>[6.x]倍增</h2><ul>
<li>ST表</li>
</ul>
<h2 id="7-x-贪心"><a href="#7-x-贪心" class="headerlink" title="[7.x]贪心"></a>[7.x]贪心</h2><link rel="stylesheet" href="/blog/css/spoiler.css" type="text/css"><script src="/blog/js/spoiler.js" type="text/javascript" async></script>
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